Ollie Pope, not out on 81, and Joe Root (55no) shared an undefeated 132-run partnership for the third wicket.
England are chasing a winning target of 296 after dismissing the world Test champions for 326 with spinner Jack Leach taking 10 wickets for the match at Headingley.
With the emphasis now on entertaining and aggressive cricket under new captain Ben Stokes, England successfully chased down 299 with some 22 overs to spare in the second Test, and they still have a full day to win the third.
"I want to be contributing to wins and I'm really enjoying working with Stokesy," Leach said.
"It's really attacking and I'm enjoying that, and kind of trying to bowl attacking as well.
"Obviously we have to come back tomorrow and do the good things again."
Earlier, Tom Blundell ran out of partners and was 88 not out at the end of New Zealand's second innings. He shared another impressive partnership with Daryl Mitchell (56).
"It's obviously disappointing: as a group we probably haven't been quite there," Blundell said.
"On a personal note, it's been fantastic to bat with Daryl and have those partnerships, it's just unfortunate that a couple of results haven't really gone our way."
The tourists resumed the fourth day on 5-168, stretching their lead to 223 at lunch.
Mitchell and Blundell have been constant irritants to England over the course of the series, sharing big stands of 195 and 236 at Lord's and Trent Bridge, respectively, and the pair once again combined to shut down the home attack.
Mitchell became the first touring player to score more than 500 runs in a Test series of three matches or fewer in England.
England started the day with an unexpected personnel change when wicketkeeper Ben Foakes Tested positive for COVID-19 and Sam Billings was summoned as an emergency replacement.
England had an early setback when Alex Lees was run out for nine.
Root arrived at 2-51 after Zak Crawley (25) was dismissed and managed to shovel a delivery from Neil Wagner over third man for six with a remarkably composed reverse scoop. The two men briefly exchanged smiles afterward.
The runs kept flowing at a rate more suited to ODIs, with twin half-centuries for Pope and Root as England hurtled along at 4.69 an over.
"The wicket is deteriorating. A little bit of variable bounce, obviously with the spin as well," Blundell added.
"It's quite hard to drive with that older ball. It looks like it's going to deteriorate even more and hopefully we can utilise that tomorrow."